Multidimensional Chebyshev spaces, hierarchy of infinite-dimensional spaces and Kolmogorov-Gelfand widths

TL;DR

This paper introduces multidimensional Chebyshev spaces and a hierarchy of infinite-dimensional spaces, extending Kolmogorov widths to multidimensional domains, with implications for signal processing and function approximation.

Contribution

It develops a multidimensional generalization of Chebyshev systems, introduces a hierarchy of infinite-dimensional spaces, and computes Kolmogorov widths for specific multidimensional function sets.

Findings

Introduced multidimensional Chebyshev spaces based on elliptic equations.

Defined a hierarchy of infinite-dimensional function spaces.

Computed Kolmogorov widths for ellipsoidal function sets in multiple dimensions.

Abstract

Recently the theory of widths of Kolmogorov (especially of Gelfand widths) has received a great deal of interest due to its close relationship with the newly born area of Compressed Sensing. It has been realized that widths reflect properly the sparsity of the data in Signal Processing. However fundamental problems of the theory of widths in multidimensional Theory of Functions remain untouched, and their progress will have a major impact over analogous problems in the theory of multidimensional Signal Analysis. The present paper has three major contributions: 1. We solve the longstanding problem of finding multidimensional generalization of the Chebyshev systems: we introduce Multidimensional Chebyshev spaces, based on solutions of higher order elliptic equation, as a generalization of the one-dimensional Chebyshev systems, more precisely of the ECT--systems. 2. Based on that we introduce a new hierarchy of infinite-dimensional spaces for functions defined in multidimensional domains; we define corresponding generalization of Kolmogorov's widths. 3. We generalize the original results of Kolmogorov by computing the widths for special "ellipsoidal" sets of functions defined in multidimensional domains.